An analytic version of stable arithmetic regularity

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary function $f(x\cdot y)$ is stable in the sense of continuous logic. Roughly speaking, our main result says that if $G$ is amenable, then any stable function on $G$ is almost constant on all translates of a unitary Bohr neighborhood in $G$ of bounded complexity. The proof uses ingredients from topological dynamics and continuous model theory. We also prove several applications which generalize results in arithmetic combinatorics to nonabelian groups.